1973 AMC 12 第 25 题

Question

一块直径为

12

英尺的圆形草地被一条宽

3

英尺的笔直碎石路切过，其中一条路边经过草地中心。剩余草地的面积（平方英尺）为

A circular grass plot

12

feet in diameter is cut by a straight gravel path

3

feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is

36\pi-34

30\pi-15

36\pi-33

35\pi-9\sqrt3

30\pi-9\sqrt3

答案：E

知识点：圆面积扇形面积分割特殊直角三角形

难度评级：2230

小提示：

用与小路垂直的圆直径将小路平分

Bisect the path by the perpendicular diameter of the circular plot

大提示：

小路的每一半由一个

30^\circ

扇形和一个

30^\circ

-

60^\circ

-

90^\circ

三角形组成

Each half of the path is a

30^\circ

sector together with a

30^\circ

-

60^\circ

-

90^\circ

triangle

解答：

草地半径为

6

。在小路的另一条边上，圆心到该边的垂直距离为

3

，所以通向交点的半径与经过圆心的路边成

30^\circ

角。小路的一半由半径为

6

的

30^\circ

扇形和直角边长为

3

与

3\sqrt3

的直角三角形组成。其面积为

\begin{aligned} &\frac{30}{360}\pi(6^2) +\frac12(3)(3\sqrt3)\\ &\qquad=3\pi+\frac{9\sqrt3}{2} \end{aligned}\text{。}

因此整条小路的面积为

6\pi+9\sqrt3

，剩余草地面积为

36\pi-(6\pi+9\sqrt3) =30\pi-9\sqrt3\text{。}

所以正确答案是 E。

The plot has radius

6.

At the path’s other edge, the perpendicular distance from the center is

3,

so the radius to an intersection point makes a

30^\circ

angle with the edge through the center. Half of the path consists of a

30^\circ

sector of radius

6

and a right triangle with legs

3

and

3\sqrt3.

Its area is

\begin{aligned} &\frac{30}{360}\pi(6^2) +\frac12(3)(3\sqrt3)\\ &\qquad=3\pi+\frac{9\sqrt3}{2}. \end{aligned}

Thus the whole path has area

6\pi+9\sqrt3,

and the remaining grass area is

36\pi-(6\pi+9\sqrt3) =30\pi-9\sqrt3.

Therefore, the correct answer is E.

1973 AMC 12 第 25 题

答案：E

难度评级：2230

小提示：

大提示：

解答：

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